Gwen's Gift

Gwen loves most numbers. In fact, she loves every number
that is *not* a multiple of $n$ (she really hates the number
$n$). For her friends’
birthdays this year, Gwen has decided to draw each of them a
sequence of $n-1$ flowers.
Each of the flowers will contain between $1$ and $n-1$ flower petals (inclusive).
Because of her hatred of multiples of $n$, the total number of petals in any
non-empty contiguous subsequence of flowers cannot be a
multiple of $n$. For
example, if $n = 5$, then
the top two paintings are valid, while the bottom painting is
not valid since the second, third and fourth flowers have a
total of $10$ petals. (The
top two images are Sample Input $3$ and $4$.)

Gwen wants her paintings to be unique, so no two paintings will have the same sequence of flowers. To keep track of this, Gwen recorded each painting as a sequence of $n-1$ numbers specifying the number of petals in each flower from left to right. She has written down all valid sequences of length $n-1$ in lexicographical order. A sequence $a_1,a_2,\dots , a_{n-1}$ is lexicographically smaller than $b_1, b_2, \dots , b_{n-1}$ if there exists an index $k$ such that $a_ i = b_ i$ for $i < k$ and $a_ k < b_ k$.

What is the $k$th sequence on Gwen’s list?

The input consists of a single line containing two integers $n$ ($2 \leq n \leq 1\, 000$), which is Gwen’s hated number, and $k$ ($1 \leq k \leq 10^{18}$), which is the index of the valid sequence in question if all valid sequences were ordered lexicographically. It is guaranteed that there exist at least $k$ valid sequences for this value of $n$.

Display the $k$th sequence on Gwen’s list.

Sample Input 1 | Sample Output 1 |
---|---|

4 3 |
2 1 2 |

Sample Input 2 | Sample Output 2 |
---|---|

2 1 |
1 |

Sample Input 3 | Sample Output 3 |
---|---|

5 22 |
4 3 4 2 |

Sample Input 4 | Sample Output 4 |
---|---|

5 16 |
3 3 3 3 |